I was given a a problem to solve, I thought I solved it but my answers don't look like the ones provided.
A cone with radius of base
r and height
h, is stretched to the left and right by length
b respectively such that height of the stretched cones equals to the height of the original cone and apexes of all the three cones are colinear.
Devise an algorithm to calculate the volume of the union of the two stretched cones (colored blue in the image).
My first question was: what is the volume of a slant cone? What I've found so far is that the formula for the volume of a slant cone is the same as that of a regular cone (which made sense to me).
So I though the answer would be 2 times the area of a cone given by radius
r and height
h minus the intersection of the two slant cones, which is also a cone with radius
r. I just need to find out the height of that smaller cone.
My thought was that I could just take the cross section of of the bodies and handle the problem in 2D. Where the base of the cone will lie on the
x axis and the center for the circle lies in point
What I did was calculate the equation of the line that goes from
(r,0). And I would also calculate the equation of the line going from
I proceeded to calculate the line equations and then to calculate where they intersect. I would then take
y value at intersection and use it as the height of the smaller cone.
I really thought I nailed it since when I had a = 0 and b = 0 I got the volume of the original cone. However, when the answers where provided mine didn't match the ones given. Can someone point out the hole in my logic?